Why does the Unit circle have a radius of 1

Question

Why is the radius of a unit circle 1? Why can't it be any other number like 10 or 50?

Answer 1

This is simply a definition of a word. The specific term "unit circle" is defined to mean the circle with radius 1.

Why might we want such a word? Well, for one thing, the unit circle can be considered as an instrument to define radian measure: the radian measure of an angle can be described as the arc length of the piece cut out of the unit circle when you draw that angle with its vertex at the center and its rays suitably extended to pass through the circle's circumference. For another, because all circles are similar, any other circle can be described as a rescaling of the unit circle, and the rescaling factor required is equal to the radius of the circle we are describing.

Answer 2

This is by definition of the word "unit". "Unit" means "one/single/individual". Hence by definition, "unit circle" is taken to mean "a circle of radius $1$".

So there's nothing deep going on mathematically; it's just a matter of how we use our English words/a manner of speaking.

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A similar question is "why do pentagons have $5$ sides"? Well, the answer is just "by definition", because in English, the prefix "pent/penta" means "five". Hence, we use "pentagon" to refer to a figure with $5$ sides.